In recent years there has been an emergence of an assortment of imaging modalities for noninvasively studying the brain. Among these, functional magnetic resonance imaging (fMRI)

^{1}^{–}^{4} and diffuse optical tomography (DOT)

^{5}^{–}^{8} are two techniques that have been developed largely in parallel to study cerebral functional hemodynamic responses. While both of these technologies are being applied successfully to a wide range of similar neuroscience and clinical topics, there are intrinsic limitations to each method, which are imposed by the governing physics of each technology (reviewed in Refs.

^{9} and

^{10}). For example, while fMRI techniques such as blood oxygen level dependent (BOLD) can provide a measurement of blood oxygen saturation changes with fairly high spatial resolution (typically 2 to 4 mm for functional studies), these signals are physiologically ambiguous, owing to the indirect relationship between changes in the transverse relaxation rate of hydrogen nuclei

and physiological hemodynamic parameters (i.e., deoxyhemoglobin and blood oxygen saturation) (reviewed in Ref.

^{11}). Although such ambiguity does not impede the use of BOLD for mapping the spatial patterns of evoked changes, this does limit the use of BOLD to directly relate physiological parameters between subjects without additional calibration methods. Calibration of the BOLD signal is possible by inducing isometabolic changes in cerebral blood flow using hypercapnia or similar vasoactive agents.

^{12}^{–}^{18} However, these hypercapnic-calibration methods require the subject to inhale increased levels of carbon dioxide gas for prolonged periods of time (up to several minutes). This procedure is both technically challenging and subject to several possible sources of systematic error

^{19} that may render the technique difficult to translate to clinical applications. While the use of hypercapnia-calibrated fMRI techniques to provide quantitative measurements of blood oxygen saturation changes has been important in applying MR techniques to study metabolism, an alternative to hypercapnia calibration is needed to make the estimation of functional CMRO

_{2} changes more routine. As CMRO

_{2} is more directly related to neural-metabolic coupling, these measurements could have significant impact in better understanding the connections between neural and hemodynamic function in health and disease (reviewed in Ref.

^{20}).

Continuous wave (cw)-based DOT has several complementary features to fMRI methods, including the ability to record a spectroscopic measurement of both oxygenated (oxyhemoglobin, HbO_{2}) and deoxygenated (deoxyhemoglobin, HbR) forms of hemoglobin. In comparison to fMRI, optical methods generally have very high temporal resolutions, with acquisition rates capable of more than several hundred hertz. This resolution is much faster than needed to capture the typical slow evoked responses and fast enough to prevent aliasing of systemic physiological signals, such as cardiac pulsation and other physiology, which can be a major source of noise in fMRI studies due to undersampling.^{21}^{,}^{22} A drawback of the DOT technology is its lower spatial resolution, which is intrinsically limited by the propagation of photons through highly scattering biological tissue (reviewed in Ref. ^{23}) and by the typically low number of optical measurement pairs recorded. Although DOT has the theoretical potential to provide quantitatively accurate measurements of hemoglobin concentration changes in the brain, in practice this can seldom be achieved because of the partial-volume effects introduced by the low spatial resolution and depth sensitivity of this method. In addition, the tomographic reconstruction of hemoglobin changes from optical measurements is generally an underdetermined and ill-posed inverse problem.^{7} Tomographic images can be improved with a greater number of measurement combinations, including overlapping measurements to provide more uniform sensitivities;^{24}^{,}^{25} however, regularization schemes must still be used to constrain the image reconstructions of the underlying absorption changes. In general, the accuracy of these reconstructions depends on the method and amount of regularization applied. In recent years, a great deal of attention has been given to this topic (reviewed in Ref. ^{26}); however, more work is still needed. One promising approach—the incorporation of prior knowledge of the spatial location of the hemodynamic change by either anatomical-based^{8}^{,}^{27}^{–}^{29} or functionally-based priors^{30}— improves the quantitative ability of DOT by constraining the solutions to the image reconstruction problem, and thus minimizing the errors introduced by partial-volume effects. With respect to optical imaging of the brain, the use of functional MRI data as such a statistical prior for the location of brain activation area has been suggested to improve DOT reconstructions.^{31} While it is believed that the introduction of statistical priors from structural or functional MRI may improve the localization of the optical signal, the implementation of such methods still has several unresolved issues. In particular, regularized reconstructions require a choice for the proper weight of the prior, as recently reviewed in Gibson, Hebden, and Acridge.^{26} In one extreme, the use of a strict (hard) prior (e.g., Ref. ^{32}) will produce images with identical spatial resolution as the original prior (e.g., the functional MR image). However, this constraint assumes that the value, location, and boundaries of the prior have negligible uncertainty. Although the signal quality of fMRI images has greatly improved in recent years due to advancements in pulse-sequence design, RF coil design, and a move to higher magnetic field strengths, background physiology, intertrial variability, and other subject-related factors are still non-negligible sources of error in these measurements and will contribute to uncertainty in a fMRI-based prior. On the other hand, the use of a statistical prior (e.g., Refs. ^{30} and ^{33}), while favorable in respect to the inclusion of the statistical uncertainty about the prior MRI information, requires knowledge of the proper statistical weight for the constraint. The optimal choice of this weighting depends on the relative measurement noise in both the fMRI and optical signals, and requires a proper statistical model of measurement noise. Concurrent multimodal measurements are unique in that physiological noise (for example, intertrial variability of the evoked response) is simultaneously recorded by each modality, while measurement noise is usually independent between instruments. This property of concurrent measurements provides an opportunity to use mutual information within multimodal measurements to help define the optimal statistical weighting of each modality in a joint image reconstruction. This concept of a bottom-up data fusion model has been previously introduced for neural imaging methods such as multimodal electroencephalography (EEG) and magnetoencephalography (MEG),^{34}^{,}^{35} but has not yet been demonstrated for multimodal hemodynamic measurements or optical methods. In this work, we describe a new analysis method for fusion of simultaneously acquired DOT and BOLD data that provides a joint estimate of the underlying physiological contrast giving rise to the concurrent measurements from both modalities. This approach makes use of the statistical properties of concurrent measurements and the commonality of the underlying physiology and fluctuations giving rise to these measurements. We use a Bayesian framework to jointly estimate brain activation changes from MR and optical using a single image reconstruction step. This approach enables us estimate oxy- and deoxyhemoglobin changes in the brain, with better spatial accuracy than DOT image reconstructions alone through the incorporation of time-varying spatial information from BOLD observations. Because the fMRI information constrains the spatial extent of the reconstruction, this helps to correct partial volume errors associated with optical reconstructions alone. Likewise, the spectroscopic information of the optical data defines the deoxyhemoglobin calibration of the BOLD signal. We find that the resulting fusion images contain quantitative information about micromolar changes in hemoglobin based on the cross-calibration of these two modalities.

We first present numerical simulations to examine the quantitative accuracy of hemoglobin estimates by our data fusion methods. Next, we apply the model to experimental data recorded simultaneously with DOT and BOLD imaging during a 2-s duration finger-walking task in five subjects.